Series : Uniform Convergence and No Zeros

Fix R>0. Show that, if n is large enough, then P_n(z)=1+z+z^2/2!+z^3/3!+...+z^n/n! has no zeros in {z:|z|<=R}

Holomorphic Function and Taylor Expansion

Let f(z) be holomorphic in |z|less than R with Taylor expansion f(z)=sum(a_nz^n) and set I_2(r)=1/2pi(integral from 0 to 2pi of|f(re^itheta)|^2 d(theta), where 0<=r

Poisson Kernel and Harmonic Function

Let P_r(t)=R((1+z)/(1-z)), z=re^it be the Poisson kernel for the unit disc |z|<1. Let U(theta) be a continous function of the interval [0,pi] with U(0)=U(pi)=0. Show that the function u(re^itheta)=1/2pi(integral from 0 to pi of {P_r(t-theta)-P_r(t+theta)}U(t)dt is harmonic in the half-disc {re^itheta,0<=r<1, 0<=theta<=pi} and ...continues

Differential Equations Proof

If y = XsinX prove that Y (X^2 + 2) - 2X dy/dx + X^2 d2y/dx2 = 0

Solving Complex Equations

I need to solve for all of the roots of (z+1)^4 = (1-i). Any idea on how to do it? keywords: imaginary

Solving Complex Variable Equations : DeMoivre's Theorem

Find the values of: i^(Sqrt(3)) The answer is: cos(pi*sqrt(3)[1/2+2k]) + i.sin(pi*sqrt(3)[1/2+2k]), for any integer k. keywords: de moivres, de moivre's

Solving Complex Equations

Please see the attached file for the fully formatted problems.

Verify a Trigonometric Identity

Prove that sin2 x + cos2 x = 1 using only this information : Cos x = (e^jx + e^-jx)/2, sin x = (e^jx – e^-jx)/(2j) See attached file for full problem description.

Complex Variables : Rectafiable Path

Fix w=re^iθ≠0 and let gamma be a rectafiable path C-{0} from 1 to w. Show there is an integer k such that ∫gamma z^-1 dz = log r + iθ + 2 pi i k See attached file for full problem description.

Complex Variables : Analytic Functions and Laurent Series

Any insight on where to go with these problems would be helpful and appreciative. This is my first time in complex analysis and I am having problems understanding the concepts. Seeing solutions to problems is helping me to understand how to approach other problems. Thank you very much! I need help specifically on problems ...continues